Questions on the use of algebraic techniques to prove geometric theorems.

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Line $mx + ny = 3$ is normal to the hyperbola $x^2 – y^2 = 1$

If the line $mx + ny = 3$ is normal to the hyperbola $x^2 – y^2 = 1$, then evaluate $\frac{1}{m^2}+\frac{1}{n^2}$. I compared given equation of normal to equation of normal at parametric point i.e ...
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2answers
25 views

The number of circles passing through the vertices of a triangle

I have read a book written by C.V Durell on Geometry. In this book I have found a lemma which states that there is one and only one circle that passes through three vertices of a triangle. I thought ...
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1answer
27 views

Draw a plane through a line parallel to the $x$-axis

Can someone help me with this problem? I bet it will be pretty easy for the most of you: Through the line $p$ draw a plane that is parallel to the $x$-axis, where p is defined by: $x=5-2t, ...
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1answer
22 views

Equation of plane passing through intersection of line and plane

Find the equation of the plane passing through the intersection of line $$\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{2}$$ and the plane $$x-y+z=5$$ and parallel to a vector with direction ratios ...
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3answers
27 views

Show that the equation of the tangent to the parabola $y^2=4ax$ at the point (p,q) is $qy=2a(x+p)$

Question: Show that the equation of the tangent to the parabola $y^2=4ax$ at the point (p,q) is $qy=2a(x+p)$ These are my two approaches: First approach: If we have $(p,q)$ as ...
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1answer
22 views

Equation of an $(n-2)-$sphere in $\mathbb{R}^n$.

I am looking for the equation of an $(n-2)$-sphere in $\mathbb{R}^{n}$ generated from the intersection of the $(n-1)$- sphere $x_1^2 + x_2^2 + \cdots + x_n^2 = r^2$, and the hyperplane perpendicular ...
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1answer
24 views

Is there any book online that shows mathematical procedures relating to perspective drawing?

I am trying to learn comprehensive mathematical analyses (rather than geometrical methods) about perspective drawing projections. Can anyone suggest a good online book to buy that illustrates ...
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2answers
83 views

Find area bounded by parabola $y^2=2px,p\in\mathbb R$ and normal to parabola that closes an angle $\alpha=\frac{3\pi}{4}$ with the positive $Ox$ axis.

Let $p=-2<0\Rightarrow y=\sqrt{-4x} \lor y=-\sqrt{-4x}\Rightarrow x\le 0 $. Let $p=2>0\Rightarrow y=\sqrt{4x} \lor y=-\sqrt{4x}\Rightarrow x\ge 0 $. For $p>0$ we can find the equation for ...
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3answers
49 views

Equations of the line which intersects the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$

Find the equations of the line which intersects the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$ and passes through the point $(1,1,1)$. First I ...
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0answers
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Find functions $\xi_1(x)$ and $\xi_2(x)$ and scalars $\alpha, \beta \in \mathbb{R}$ to characterize a set

Find functions $\xi_1(x)$ and $\xi_2(x)$ and scalars $\alpha, \beta \in \mathbb{R}$, such that $\xi_1(x) \le y \le \xi_2(x)$ when $\alpha \le x\le \beta$ equivalent to to the following set in ...
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1answer
31 views

lattice point on a circle

consider a circle with center (sqrt[2],1/3) and any arbitrary radius. how do I prove that there is atmost one lattice point on the circle? also, does there exist an unique cirle with exactly 2004 ...
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0answers
13 views

Geometric interpretation of zeros of differential of complex polynomial

I want to show that if three complex numbers $ a,b,c$ don't lie on the same line, then if $p,q$ are such that $W^{'}(p)=0=W^{'}(q)$ where $W(z)=(z-a)(z-b)(z-c)$ then angles $acp,bcq$ are equal.I have ...
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3answers
71 views

Curve-fitting using circles

I'm working for a firm, who can only use straight lines and (parts of) circles. Now I would like to do the following: imagine a square of size $5\times5$. I would like to expand it with $2$ in the ...
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2answers
24 views

Determine whether two segments P1Q1 and P2Q2 have a common point if the (x,y) coordinates of their end points is known?

Does this question have a solution? I think it's impossible to know if line segments P1Q1 and P2Q2 intersect at all with just the information about their end points Q1 and Q2. Thanks.
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3answers
74 views

Reflection of a Light Ray

I found this problem to be very hard while studying for the exam: Let $$L: \vec r(t)=<1,-2,3>+t<-5,4,1>, \qquad t \in \mathbb{R}$$ be a line. Light is traveling along the line $L$ ...
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1answer
52 views

Coordinate Geometry finding x and y

How would I rearrange this equation to find $x_3$ and $y_3$ $$\tan\ \alpha =\frac{\sqrt{(x_3-x_2)^2+(y_3-y_2)^2}}{ \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}$$ EDIT: So basically what I want to do is that I ...
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1answer
18 views

Limiting points subtend right angle at the centre

If the limiting points of the system of circles $x^2+ y^2+ 2gx +w(x^2+ y^2+ 2fy + k)=0$ where $w$ is a parameter , subtends a right angle at origin then find value $k/f^2$? I know that limiting point ...
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1answer
52 views

Three circles intersect at one point.

If three circles intersect at one point then there's unique $x$ and $y$ coordinate values such that the following equations are satisfied: $$(x-x_i)^2 + (y-y_i)^2 = r_i^2$$ Where $i=1,2,3$ Taking ...
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1answer
43 views

Relation between $a$ and $b$ when equation of obtuse angle bisector is $ax+by-3=0$

The combined equation of bisector of angles between the lines $L_1$ and $L_2$ is $$2x^2-3xy-2y^2-x+7y-3=0$$ $P(4,-3)$ is a point on $L_1$. If the equation of obtuse angle bisector is ...
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1answer
38 views

Systems of Equations (Inconsistent)

Question: Consider the following system of three equations: $$2y+2z=9-2x$$ $$x=12-3y-4z$$ $$Ax+5y+6z=B$$ Find values of A and B which makes the system of equations ...
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1answer
30 views

Systems Of Equations (Find any other equation in the form $ax + by + cz = k $)

Question: Prove the point $(2,5,-4)$ is a solution to the two equations: $$ x + 2y + 3z = 0 $$ $$ 2x-y-2z=7$$ Find any other equation in the form $$ax + by + cz = k $$ ...
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2answers
31 views

Is the intersection of two wedge-shaped regions also wedge-shaped?

In the plane $\mathbb{R}^2$, the intersection of two wedge-shaped regions should still be wedge-shaped. However I don't see where to go from here. I searched for wedge-shaped and couldn't find ...
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1answer
16 views

$y^2 = 2a(x+a\sin \frac{x}{a})$ and tangents parallel to $x$ axis

Prove that all the points on the curve $$y^2 = 2a(x+a\sin \frac{x}{a})$$ at which tangent is parallel to the axis of $x$, lie on a parabola. Here slope of tangent at $(h,k)$ must be $0$. After ...
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1answer
32 views

Minimization vs. Maximization of the area of a triangle inscribed in a triangle

We’ve been given a triangle ABC with an area = $1$. Now Marcus gets to choose a point $X$ on the line $BC$, afterwards Ashley gets to choose a point $Y$ on line $CA$ and finally Marcus gets to choose ...
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3answers
134 views

Valid alternative to L'Hopital?

This question concerns a proof of the area formula for the unit circle: $A = \pi$. Start with the area formula for a unit polygon: $$A = \frac{1}{2}n\sin{\frac{2\pi}{n}}$$ The proof then goes on to ...
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3answers
77 views

Is this a coincidence?

I was solving a problem and I noticed something which was very curious, the problem itself was easy: Find the point of intersection of the lines: $$2x+3y=5$$ and $$(1+c)x+(2+c)y=4+c$$ when c ...
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2answers
30 views

Given the points $A(3, 2)$ and $B(-5,-3)$, what is the product of the coordinates of the midpoint of $\overline{AB}$?

Given the points $A(3, 2)$ and $B(-5, -3)$, what is the product of the coordinates of the midpoint of $\overline{AB}$? Express your answer as a common fraction. I've tried to find the length of ...
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2answers
26 views

Parametrization of a lemniscate

I wonder how to parametrize lemniscate in the following way: $\gamma : (-\infty, \infty) \rightarrow \mathbb R^2,$whose image is a lemniscate whose axes of symmetry are $y=x$ and $y+x=0$. The ...
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2answers
20 views

Finding the Equations of a Circle provided a point and the radius

So I tried googling the exact question and I never found the solution. This is homework so I really don't want to know the answer but how to arrive at the answer. The question that was given was ...
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2answers
31 views

Family of circles with $AB$ as diameter

The circle $S_1 :x^2+y^2-4=0$ cuts the circle $S_2 :x^2+y^2+2x+3y-5=0$ in A and B. Then find the equation of circle with $AB$ as diameter. Answer is $13(x^2+y^2)-4x-6y-50=0$ Equation of AB will ...
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1answer
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Equation of circumcircle formed by $x^2+y^2+2gx+2fy=0$ and $2x+y=1$

​​​The equation of circumcircle of triangle formed by lines $7x^2+8xy-y^2=0$ and $2x+y=1$ is $x^2+y^2+2gx+2fy=0$ ,then find $g$ and $f$ I thought if I make equation of circle homogeneous with the ...
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1answer
28 views

A problem in analytic geometry

Given points A and B, and a line p with its equation $p:\vec{r}=\vec{r_{p}}+t\vec{p}, t\in R$ such that $\vec{p}$ is not parallel to $\vec{AB}$. Find points C and D, as a function of $\vec{r_{a}}, ...
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0answers
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In $\triangle ABC, PA+PB+PC=AB+AC, PE=x, f(x)=x+BE+CE, $then $f(x)$ is mono increasing function

In $\triangle ABC, PA+PB+PC=AB+AC, PA$ extend to cross $BC$ at $D, E$ is on $PD$, let $PE=x,f(x)=x+EB+EC$, then $f(x)$ is mono increasing function. I can write $BE=\sqrt{AB^2+(AP+x)^2-2AB*(AP+x) ...
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1answer
150 views

How to calculate volume of a right circular cone's hyperbola segment?

PROBLEM I am working on calculating volumes of geometric solids. All shapes have been pretty basic until now. I am bewildered on how to attack the problem of calculating the volume of a slice of a ...
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1answer
27 views

Calculate rectangle's side length from fitted ellipse and area data

The problem is the following: Let there be a rectangle with sides $a$ and $b$ and diagonal $d$. Let there be an ellipse with axes $x$ and $y$. The areas of the two shapes are the same. The ...
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0answers
21 views

I need to find the diameter of an inscribed circle inside the union of 3 circles.

I need to find the diameter of an inscribed circle inside the union of 3 circles. This is sort of the inverse of the "problem of Apollonius" solved elsewhere. The circles aren't tangent, but form an ...
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47 views

Function that generates a Grass Leaf (as a list of vectors (or vertices))

I've tried to implement procedural grass generation into my little Graphics Engine but got stuck at the following mathematical problem: Let ...
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1answer
53 views

Equation to the circle.

How to show that the equation to the circle of which the points $(x_1,y_1)$ and $(x_2,y_2)$ are the ends of a cord of a segment containing an angle $\theta$ is, $$(x-x_1)(x-x_2)+(y-y_1)(y-y_2) ± ...
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1answer
29 views

How to derive the volume of a tetrahedron with the following data? [closed]

The vertices of a tetrahedron are:- A - (0, 0, 0) B - (0, 0, a) C - (0, b, 0) D - (c, 0, 0) Prove that the volume is:- 1/6 abc. A figure will be helpful.
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2answers
82 views

Are planes in $3$-dimensions two-dimensional?

Are planes in $3$-dimensions two-dimensional? The reason I ask is because mathematically the $xy$-plane exists in $3$D space but appears to be $2$D, but how can something $2$D be in $3$D space? I ...
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1answer
32 views

If two monic polynomials have no common roots, are the coefficients of their product locally diffeomorphic to the product of the coefficients?

Let $P^d (t,\lambda)$ be the "generic" d-th degree monic polynomial $P^d (t,\lambda) = t^d + \sum\limits_{i=1}^d \lambda_i t^{d-i}$ with real coefficients. Let $\lambda(\xi,\eta)$ be given by the ...
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2answers
25 views

Find the equation of the line which is

Find the equation of the line perpendicular to the line joining the points $A(3,6)$ and $B(-6,9)$, which divides the line $AB$ in the ratio of $2:1$. My attempt: Equation of $AB$ is ...
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1answer
50 views

How do I find the difference between the gradients of two lines represented by an equation

I want to find the difference between the gradients (or slopes?) of two lines. The equation of the lines is $$x^2(\tan^2 \theta+\cos^2 \theta)-2xy\tan\theta+y^2 \sin^2 \theta=0$$ I have assumed the ...
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1answer
69 views

prove that the quadrilateral $ABCD$ is a square

Given $ABCD$ a quadrilateral such that $AB\parallel CD$ and $\angle ACD=45^0, \angle A=90^0, \angle D=90^0 $ Need to prove that $ABCD$ is a square. I tried to use circles but it didn't help. Any ...
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3answers
55 views

curve represented by the given equation. [closed]

What does the equation $$x^2y-2xy-3y^2-4x^2+8x+12=0$$ represent. Im blank on it dont know how to proceed on it as all the equation seems to be quadratic in $x,y$
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3answers
54 views

The line $2x-y=5$ turns about a point…

The line $2x-y=5$ turns about a point on it, whose ordinate and abscissae are equal, through an angle of $45°$, in anti clockwise direction. Find the equation of line in the new position. My attempt ...
2
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3answers
55 views

How to find position of point that is x unit distant from AB line segment and y unit distant from BC line segment?

I am trying to calculate coordinates of point P, which is x units distant from AB line segment and y units distant from BC line segment. Edit: I am trying to write code for general solution. As ...
1
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1answer
22 views

Find the equation of the radius

Find the equation of the radius of the circle $x^2-4x-6y+y^2=23$ and passing through the point $(4,5)$. My attempt: Here the equation of the circle is: $$x^2-4x-6y+y^2=23$$ $$(x-2)^2+(y-3)^2=6^2$$ ...
2
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2answers
114 views

Intersection of cone and cylinder layout formula for sheet metal application

A common part in HVAC is a cylindrical pipe intersecting a truncated cone. I am designing a machine to mass produce this part. I would cut the parts out of sheet metal and roll them up to form the ...
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5answers
24 views

Prove that one of the lines represented by $ax^2+2hxy+by^2=0$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2$.

Prove that one of the lines represented by $ax^2+2hxy+by^2=0$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2$. Solution I calculated the two lines represented by ...